Connections within the group of hyperbolic functions and with other function groups

Representations through more general functions

The hyperbolic functions are particular cases of more general functions. Among these more general functions, four classes of special functions are of special relevance: Bessel, Jacobi, Mathieu, and hypergeometric functions.

For example, and have the following representations through Bessel, Mathieu, and hypergeometric functions:

All hyperbolic functions can be represented as degenerate cases of the corresponding doubly periodic Jacobi elliptic functions when their second parameter is equal to or :

Representations through related equivalent functions

Each of the six hyperbolic functions can be represented through the corresponding trigonometric function:

Relations to inverse functions

Each of the six hyperbolic functions is connected with a corresponding inverse hyperbolic function by two formulas. One direction can be expressed through a simple formula, but the other direction is much more complicated because of the multivalued nature of the inverse function:

Representations through other hyperbolic functions

Each of the six hyperbolic functions can be represented through any other function as a rational function of that function with a linear argument. For example, the hyperbolic sine can be representative as a group‐defining function because the other five functions can be expressed as:

All six hyperbolic functions can be transformed into any other function of the group of hyperbolic functions if the argument is replaced by with :